document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201132>Question 1201132</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Formulate a system of equations for the situation below and solve.\r<BR>\n" );
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document.write( "A theater has a seating capacity of 750 and charges $4 for children, $6 for students, and $8 for adults. At a certain screening with full attendance, there were half as many adults as children and students combined. The receipts totaled $4600. How many children attended the show?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #835379 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=835379\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>200 children</font>\r<BR>\n" );
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document.write( "Work Shown:\r<BR>\n" );
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document.write( "c = number of children<BR>\n" );
document.write( "s = number of students<BR>\n" );
document.write( "a = number of adults<BR>\n" );
document.write( "These are nonnegative integers  {0,1,2,3,...}\r<BR>\n" );
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document.write( "\"<font color=blue>A theater has a seating capacity of 750 ... At a certain screening with full attendance</font>\"<BR>\n" );
document.write( "c+s+a = total number of people<BR>\n" );
document.write( "c+s+a = total seating capacity = 750<BR>\n" );
document.write( "c+s+a = 750<BR>\n" );
document.write( "Let's refer to this as equation (1).\r<BR>\n" );
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document.write( "4c = revenue from the children only<BR>\n" );
document.write( "6s = revenue from the students only<BR>\n" );
document.write( "8a = revenue from the adults only<BR>\n" );
document.write( "4c+6s+8a = total revenue = 4600 dollars<BR>\n" );
document.write( "4c+6s+8a = 4600<BR>\n" );
document.write( "2(2c+3s+4a) = 4600<BR>\n" );
document.write( "2c+3s+4a = 4600/2<BR>\n" );
document.write( "2c+3s+4a = 2300<BR>\n" );
document.write( "Let's refer to this as equation (2).\r<BR>\n" );
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document.write( "\"<font color=blue>there were half as many adults as children and students combined</font>\"<BR>\n" );
document.write( "c+s = number of children and students combined<BR>\n" );
document.write( "0.5(c+s) = half the previous amount = number of adults<BR>\n" );
document.write( "0.5(c+s) = a<BR>\n" );
document.write( "Let's refer to this as equation (3).\r<BR>\n" );
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document.write( "System of equations:<BR>\n" );
document.write( "c+s+a = 750<BR>\n" );
document.write( "2c+3s+4a = 2300<BR>\n" );
document.write( "0.5(c+s) = a\r<BR>\n" );
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document.write( "Let's multiply both sides of the third equation by 2, then get everything to one side.<BR>\n" );
document.write( "0.5(c+s) = a<BR>\n" );
document.write( "a = 0.5(c+s)<BR>\n" );
document.write( "2*a = 2*0.5(c+s)<BR>\n" );
document.write( "2a = c+s<BR>\n" );
document.write( "0 = c+s-2a<BR>\n" );
document.write( "c+s-2a = 0<BR>\n" );
document.write( "This is the updated version of equation (3).\r<BR>\n" );
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document.write( "Updated system of equations:<BR>\n" );
document.write( "c+s+a = 750<BR>\n" );
document.write( "2c+3s+4a = 2300<BR>\n" );
document.write( "c+s-2a = 0\r<BR>\n" );
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document.write( "Narrow your focus to equations (1) and (3)<BR>\n" );
document.write( "c+s+a = 750<BR>\n" );
document.write( "c+s-2a = 0<BR>\n" );
document.write( "Both start with <font color=blue>c+s</font><BR>\n" );
document.write( "Let's subtract straight down to cancel out each copy of <font color=blue>c+s</font><BR>\n" );
document.write( "More specifically...<ul><li>The c variables cancel.</li><li>The s variables cancel.</li><li>For the 'a' variables we have a - (-2a) = a+2a = 3a</li><li>For the right hand sides we have 750-0 = 750</li></ul>So,<BR>\n" );
document.write( "3a = 750 <BR>\n" );
document.write( "3a/3 = 750/3<BR>\n" );
document.write( "a = 250 is the number of adults\r<BR>\n" );
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document.write( "Plug that value of 'a' into the first equation of the system. <BR>\n" );
document.write( "Then isolate c.<BR>\n" );
document.write( "c+s+a = 750<BR>\n" );
document.write( "c+s+250 = 750<BR>\n" );
document.write( "c+s = 750-250<BR>\n" );
document.write( "c+s = 500<BR>\n" );
document.write( "c = 500-s<BR>\n" );
document.write( "We'll come back to this later.\r<BR>\n" );
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document.write( "Revisit the second equation.<BR>\n" );
document.write( "Plug in a = 250 and do a bit of algebra.<BR>\n" );
document.write( "2c+3s+4a = 2300<BR>\n" );
document.write( "2c+3s+4*250 = 2300<BR>\n" );
document.write( "2c+3s+1000 = 2300<BR>\n" );
document.write( "2c+3s = 2300-1000<BR>\n" );
document.write( "2c+3s = 1300\r<BR>\n" );
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document.write( "Then plug in c = 500-s<BR>\n" );
document.write( "This will allow us to solve for s.<BR>\n" );
document.write( "2c+3s = 1300<BR>\n" );
document.write( "2(500-s)+3s = 1300 <BR>\n" );
document.write( "1000-2s+3s = 1300<BR>\n" );
document.write( "1000+s = 1300<BR>\n" );
document.write( "s = 1300-1000<BR>\n" );
document.write( "s = 300 is the number of students\r<BR>\n" );
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document.write( "We'll use that value of s to determine c.<BR>\n" );
document.write( "c = 500-s<BR>\n" );
document.write( "c = 500-300<BR>\n" );
document.write( "<font color=red>c = 200 is the number of children</font>\r<BR>\n" );
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document.write( "-------------------------------------------<BR>\n" );
document.write( "Summary\r<BR>\n" );
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document.write( "c = 200, s = 300, a = 250\r<BR>\n" );
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document.write( "<font color=red>200 children</font><BR>\n" );
document.write( "300 students<BR>\n" );
document.write( "250 adults\r<BR>\n" );
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document.write( "Check:<BR>\n" );
document.write( "c+s+a = 750<BR>\n" );
document.write( "200+300+250 = 750<BR>\n" );
document.write( "500+250 = 750<BR>\n" );
document.write( "750 = 750 ....... first equation works<BR>\n" );
document.write( "and<BR>\n" );
document.write( "2c+3s+4a = 2300<BR>\n" );
document.write( "2*200+3*300+4*250 = 2300<BR>\n" );
document.write( "400+900+1000 = 2300<BR>\n" );
document.write( "1300+1000 = 2300<BR>\n" );
document.write( "2300 = 2300 ....... second equation works<BR>\n" );
document.write( "and<BR>\n" );
document.write( "0.5(c+s) = a<BR>\n" );
document.write( "0.5(200+300) = 250<BR>\n" );
document.write( "0.5(500) = 250<BR>\n" );
document.write( "250 = 250  ....... third equation works<BR>\n" );
document.write( "All three equations of the original system are true for the c, s and 'a' values mentioned.<BR>\n" );
document.write( "Therefore, the answers are confirmed.\r<BR>\n" );
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document.write( "Other approaches are possible. Feel free to explore other options.<BR>\n" );
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